Tight Relaxations for Polynomial Optimization and Lagrange Multiplier Expressions

TL;DR

This paper introduces new tight semidefinite relaxations for polynomial optimization, leveraging polynomial expressions of Lagrange multipliers, which ensure finite convergence and improved solution accuracy.

Contribution

It develops a novel hierarchy of relaxations using polynomial Lagrange multiplier expressions, enhancing the tightness and convergence properties of polynomial optimization methods.

Findings

Hierarchy of relaxations has finite convergence.

Lagrange multipliers can be expressed as polynomial functions.

New relaxations are tight at finite relaxation order.

Abstract

This paper proposes tight semidefinite relaxations for polynomial optimization. The optimality conditions are investigated. We show that generally Lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points. The polynomial expressions can be determined by linear equations. Based on these expressions, new Lasserre type semidefinite relaxations are constructed for solving polynomial optimization. We show that the hierarchy of new relaxations has finite convergence, or equivalently, the new relaxations are tight for a finite relaxation order.